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We do? What did you think the European Parliament elections every four years were for?
> directly elect some kind of president.
Why? Nowhere in Western Europe except very arguably France (France, as always, has to be a bit weird about everything, and has a hybrid system) has a directly elected executive. True executive presidential systems are only really a thing in the Americas and Africa (plus Russia, these days).
Like, in terms of big countries with a true executive presidency, you’re basically looking at the US, Russia and Brazil. I’m, er, not sure we should be modeling ourselves on those paragons of democracy.
> They have no accountability, no checks and balances.
The parliament has the same accountability and checks and balances as any national parliament, more or less (more than some, as the ECJ is more effective and independent than many national supreme courts).